If I want to delete a file which I'm not the owner of, then I have to enter my password (I'm in the admin group). The same happens if I want to move a file into a folder with different ownership.
But there is no way to rename a file with different ownership. Although there is a workaround: copy the file to your desktop, so that the ownership is being changed. Then rename the file and move it back --> here the admin pw is needed.
Let the user rename files with different ownership the same way as normal renaming. Just ask for the password, then enable the rename mode.